Fluid Theory: Magnetohydrodynamics (MHD)

Theoretical Wave Analyses Oscillations: frequency, amplitude Waves: amplitude, frequency, wavelength, (propagation speed) at a fixed point: oscillation is time a snapshot: oscillation in space a movie shot: propagating oscillations phase velocity: following a fixed phase Eigen modes: coherent oscillation propagation intrinsic to the medium (independent of source for a frequency) dispersion relation: frequency dependence on wavelength, angle perturbation relations: among different quantities In finite space: Eigen modes can also refer to resonating frequencies

MHD Waves

MHD Waves, cont.

MHD Waves, continue Dispersion relations Phase velocity Perturbation relations Walén relation (for intermediate mode) : for k antiparallel to B and parallel to B

Alfvén mode (Intermediate Mode) In the Alfven mode, perturbations of B are perpendicular to both B and k. There is no change in density r or in the magnitude of B. The changes involve only bends of B. Alfvén mode waves work to reduce bending of the magnetic field. They carry field-aligned currents.

Fast and Slow Modes The fast (+ sign) and slow (- sign) modes that make up the other two solutions are compressional (i.e. they do change density and field magnitude). Fast waves are produced when the total pressure of the plasma (the sum of the particle pressure and field pressure) changes locally. The plasma and magnetic pressures are in phase. This wave propagates almost isotropically.

Fast and Slow Modes – cont. For the slow mode the total pressure is approximately constant across the background field. Slow mode waves carry energy primarily along the background field. Field-aligned gradients in the total pressure drive slow mode waves.

Homework 3.1, 3.7, 4.10, 5.2, 3.10*, 3.12*, 4.8* Hint, 4.10: the wave energy is the part of the energy that is associated with the wave or perturbations. Hint, 4.10: equations (4.34) and (4.35) are incorrect.

1-D discontinuities Shock Front v1 v2 Upstream (low entropy) Downstream (high entropy) v1 v2

Rankine-Hugoniot Relations

Types of Discontinuities in Ideal MHD The R-H jump conditions are a set of 6 equations. If we want to find the downstream quantities given the upstream quantities then there are 6 unknowns ( ,vn,,vT,,p,Bn,BT). The solutions to these equations are not necessarily shocks. These conservations laws and a multitude of other discontinuities can also be described by these equations. Types of Discontinuities in Ideal MHD Contact Discontinuity , Density jumps arbitrary, all others continuous. No plasma flow. Both sides flow together at vT. Tangential Discontinuity Complete separation. Plasma pressure and field change arbitrarily, but pressure balance Rotational Discontinuity Large amplitude intermediate wave, field and flow change direction but not magnitude.

Types of Shocks in Ideal MHD Shock Waves Flow crosses surface of discontinuity accompanied by compression. Parallel Shock B unchanged by shock. Perpendicular Shock P and B increase at shock Oblique Shocks Fast Shock P, and B increase, B bends away from normal Slow Shock P increases, B decreases, B bends toward normal. Intermediate Shock B rotates 1800 in shock plane, density jump in anisotropic case

Configuration of magnetic field lines for fast and slow shocks Configuration of magnetic field lines for fast and slow shocks. The lines are closer together for a fast shock, indicating that the field strength increases. [From Burgess, 1995].

For compressive fast-mode and slow-mode oblique shocks the upstream and downstream magnetic field directions and the shock normal all lie in the same plane: coplanarity theorem. The transverse component of the momentum equation can be written as and Faraday’s Law gives . Therefore both and are parallel to and thus are parallel to each other. Thus . Expanding If and must be parallel. The plane containing one of these vectors and the normal contains both the upstream and downstream fields. Since this means both and are perpendicular to the normal and

Formation of Sonic Shock